Although Euclid's Elements were written more than two thousand years ago, theorems about triangles continue to be invented. One interesting example is Euler's theorem that the barycenter, circumcenter and orthocenter of any triangle are colinear.
The line joining the vertex of a triangle to the midpoint of the opposite side is called a median. The three medians of any triangle intersect at a single point called the Barycenter, or center of gravity. The three medians appear purple in the applet below.
Finally, the Circumcenter is the center of the circle passing through all three vertices. By joining the midpoints of all three edges, we can construct a smaller triangle. Its barycenter will be the larger triangle's circumcenter. The smaller triangle and its altitudes appear yellow in the applet below.
The following illustration shows Euler's theorem. Euler's line appears orange. Try dragging the triangle's vertices with the mouse (if your mouse has more than one button, use the left one). As the three points move, they remain colinear.
David E. Joyce of Clark University has written a much better system that specializes in demonstrating geometric constructions. When I saw his system, I realized I could adapt mine to achieve a similar effect. His Geometry applet is available here.